Why isn’t P and P/poly trivially the same?

The definition of P is a language that can be decided by a polynomial time algorithm. The definition of P/poly can be taken to mean a language that can be decided by a polynomial-size circuit (see http://pages.cs.wisc.edu/~jyc/02-810notes/lecture09.pdf). Now, why can’t a polynomial-sized circuit be simulated in polynomial time?